Cracovians: The Twisted Twins of Matrices

Summary

Linear algebra is typically explained using matrices. But matrix theory is just one possible perspective. Below, I describe an alternative approach to linear algebra. Tadeusz Banachiewicz (1882–1954), a Polish astronomer living in Krakow, was passionate about calculating machines. From the 1920s, Banachiewicz developed a method for computations on tables of numbers, which was particularly easy to perform with arithmometers. In honor of Krakow, Banachiewicz named these computational objects cracovians. In the preface to Banachiewicz’s book, Cracovian Algebra (Rachunek krakowianowy, PWN 1959), it is noted that “some calculations, generally known to be very tedious, became a sheer entertainment for the executor.” Similar to matrices, cracovians are represented as rectangular tables of numbers. The equality of cracovians, addition of cracovians, and multiplication of cracovians by a scalar are defined identically to their matrix counterparts. However, the multiplication of a cracovian by another cracovian is defined differently: the result of multiplying an element from column i of the left cracovian by an element from column j of the right cracovian is a term of the sum in column i and row j of the result. In essence, each column of the left cracovian is multiplied by each column of the right cracovian. Here’s an example (cracovians are enclosed in braces to distinguish them from matrices): All such cracovians whose elements on the main diagonal are equal to 1 and other elements are zeros are denoted by the Greek letter τ (tau). Any such unit cracovian τ as the second factor of multiplication does not change the cracovian by which it is multiplied. As the first factor of multiplication, a unit cracovian τ transposes the cracovian by which it is multiplied: For any such cracovians A and B that have the same number of rows, the identity holds: Therefore, in the general case, cracovian multiplication is not commutative. By convention, we agree that cracovian multiplicatio...